Theorem ismri2dad 16297

Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ismri2dad.1	|- N = (mrCls ` A )
ismri2dad.2	|- I = (mrInd ` A )
ismri2dad.3	|- ( ph -> A e. (Moore ` X ) )
ismri2dad.4	|- ( ph -> S e. I )
ismri2dad.5	|- ( ph -> Y e. S )

Assertion

Ref	Expression
ismri2dad	|- ( ph -> -. Y e. ( N ` ( S \ { Y } ) ) )

Proof of Theorem ismri2dad

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismri2dad.4	. . 3 |- ( ph -> S e. I )
2		ismri2dad.1	. . . 4 |- N = (mrCls ` A )
3		ismri2dad.2	. . . 4 |- I = (mrInd ` A )
4		ismri2dad.3	. . . 4 |- ( ph -> A e. (Moore ` X ) )
5	3, 4, 1	mrissd 16296	. . . 4 |- ( ph -> S C_ X )
6	2, 3, 4, 5	ismri2d 16293	. . 3 |- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
7	1, 6	mpbid 222	. 2 |- ( ph -> A. x e. S -. x e. ( N ` ( S \ { x } ) ) )
8		ismri2dad.5	. . 3 |- ( ph -> Y e. S )
9		simpr 477	. . . . 5 |- ( ( ph /\ x = Y ) -> x = Y )
10	9	sneqd 4189	. . . . . . 7 |- ( ( ph /\ x = Y ) -> { x } = { Y } )
11	10	difeq2d 3728	. . . . . 6 |- ( ( ph /\ x = Y ) -> ( S \ { x } ) = ( S \ { Y } ) )
12	11	fveq2d 6195	. . . . 5 |- ( ( ph /\ x = Y ) -> ( N ` ( S \ { x } ) ) = ( N ` ( S \ { Y } ) ) )
13	9, 12	eleq12d 2695	. . . 4 |- ( ( ph /\ x = Y ) -> ( x e. ( N ` ( S \ { x } ) ) <-> Y e. ( N ` ( S \ { Y } ) ) ) )
14	13	notbid 308	. . 3 |- ( ( ph /\ x = Y ) -> ( -. x e. ( N ` ( S \ { x } ) ) <-> -. Y e. ( N ` ( S \ { Y } ) ) ) )
15	8, 14	rspcdv 3312	. 2 |- ( ph -> ( A. x e. S -. x e. ( N ` ( S \ { x } ) ) -> -. Y e. ( N ` ( S \ { Y } ) ) ) )
16	7, 15	mpd 15	1 |- ( ph -> -. Y e. ( N ` ( S \ { Y } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   {csn 4177   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246  df-mri 16248

This theorem is referenced by:  mrieqv2d  16299  mreexmrid  16303  mreexexlem2d  16305  acsfiindd  17177